Download 14(2)

COMBINATORIAL NUMBERS IN 0n
110
(9.16)
April 1976
(2 + tf + tln) = 0.
1]n. It follows that
It is shown in [1] that (9.15) can be inverted to give t(n)= [e[e+l]n
(9.17)
+ [e-1]n
= 0,
n > 0.
As before we introduce M e W(n) and r\(M) = [e(m 1), e(rr)2l - , e(mn)],
ri(n,Mf =
u
fo(M)]
5
=
with
e(mk).
m=1
The /7-dimensional tangent coefficients will be T(M) = Mm f), t(m2>, •••, t(mn)], so that
t(n,M) = [T(M)]U = 5
t(mk).
k—1
Finally let e(M) = [e(m 1), e(m2), - , e(mn)], so that
e(n,M) = [e(M)]M
n
II
=
e(mk),
k=1
where the numbers e(n,M) are called the /7-dimensional Euler numbers. It is easily seen, like in the case of the Bernoulli numbers, that
[e(n)+1]p+[e(n>-
(9.18)
1]p = 0,
p
(9.19)
t(n,P) + [2U + T(n)]
(9.20)
= 0,
P > 0,
K!2Ke(n,K),
t(n,K) =
e(n,P) = [U+T(n)]p
(9.21)
,
M
(9.22)
P > 0,
t(n,M) = [e(n)-U]
.
We introduce in the same way the /7-dimensional Euler polynomials: Let
HIP) = fo(p i,Xf),
V(P2, *2>, - , Vfon, Xn)J ,
3
where/ e W(n). It follows that
P
n
I I r\(pk,xk)
= J^ e(n,K)Xp'K/(P
k 1
~
K=0
which defines the/7-dimensional Euler polynomials.
It can easily be checked that similarly to the one-dimensional case we have
(9.23)
(9.24)
and
(9.25)
n(P,K) =
0v(P,X) =
- K)l ,
r\(P-U,X)
Mr\(P,X) = Xp/P!
.
According to Section 8 we obtain the following generating function for the Euler numbers e(n,K) and the numbers
e(n.K)
(9.26)
(9.27)
Ge(n,P) = 2/[eT
+ e~T]
Ge(n,P) = 2/[eT
= ]T
+ 1] = £
etn,K)TK/'K!
efn.KJT* .
REFERENCES
Ch. Jordan, Calc. of Finite Differences, New York, 1950.
J. Riordan, Combinatorial Identities, New York, 1968.
S. Tauber, "On Multinomial Coefficients," Am. Math. Monthly, 70(1963), 1058-1063.
S. Tauber, "On /7-dimensional Stirling Numbers,:: Proc. Edinburgh Math. Soc, 16(1969), Series II, Part4. pp.
291-299.
5. S. Tauber, "On Quasi-Orthogonal Numbers," Am. Math. Monthly, 69(1962), pp. 365-372.
1.
2.
3.
4.
*******